Convergent Series Identities

1. The problem statement, all variables and given/known data
a) If c is a number and [tex]\sum a_{n}[/tex] from n=1 to infinity is convergent to L, show that [tex]\sum ca_{n}[/tex] from n=1 to infinity is convergent to cL, using the precise definition of a sequence.

b)If [tex]\sum a_{n}[/tex] from n=1 to infinity and [tex]\sum b_{n}[/tex] from n=1 to infinity are convergent to X and Y respectively, show that [tex]\sum b_{n}+a_{n}[/tex] from n=1 to infinity is convergent to X+Y.

2. Relevant equations
I personally thought these were identities, and have no idea how to approach them.

3. The attempt at a solution
a) Maybe [tex]\sum a_{n}[/tex] from n=1 to infinity = [tex] Lim (S_{n}) [/tex] as n goes to infinity, has something to do with it

These proofs essentially rely on the definition of convergence for infinite series, i.e. that the mth partial sum converges to some limit L as m goes to infinity
Can you reformulate the question in terms of partial sums?

(Note that you can do what you'd expect to do with sums when they are only finite, but not necessarily when they are infinite)

The definition of [tex]Lim(S_{n})[/tex] is [tex]|S_{n}-L|<\epsilon\rightarrow \forall n>N[/tex] right? Can I multiply both sides of [tex]|S_{n}-L|<\epsilon[/tex] by c? But then there are two cases depending on the sign of c right?

The definition of [tex]Lim(S_{n})[/tex] is [tex]|S_{n}-L|<\epsilon\rightarrow \forall n>N[/tex] right? Can I multiply both sides of [tex]|S_{n}-L|<\epsilon[/tex] by c? But then there are two cases depending on the sign of c right?

Ok, so can I say let [tex]\epsilon_{1}=\frac{\epsilon_{2}}{|c|}[/tex] so [tex]|S_{n}-L|<\epsilon_{1}\rightarrow \frac{\epsilon_{2}}{|c|} [/tex], so [tex]|S_{n}-L|<\frac{\epsilon_{2}}{|c|} \rightarrow |cS_{n}-cL|<\epsilon_{2}[/tex]?